1

Phase Space Instability with Frequency Sweeping

H. L. Berk and D. Yu. Eremin

Institute for Fusion Studies

Presented at IAEA Workshop

Oct. 6-8 2003

2

“Signature” for Formation of Phase Space Structure (single resonance)

Explosive response leads to formation of phase space structure

Berk, Breizman, Pekker

3

Simulation:N. Petviashvili

4

“BGK” relation•Basic scaling obtained even by neglecting effect of directfield amplitude •Examine dispersion with a structure in distribution function(e.g. hole)

€

0 = ε(ω,k) =1 +ω p

2

kdv∫

∂f (v)

∂vω − kv

≈2(ω −ω p )

ω p

+ω p

2

kdv∫

∂[ f (v) − f0(v)]

∂vω − kv

0 ≈ω − ωp

ω(1−

γ L

ωb

); thus γ L ≈ ωb

v

€

ωb

k

€

f (v) - f0 (ω0

k)

5

Power Transfer by Interchange in Phase Space

Ideal Collisionless Result

€

ωb =163π 2 γ L ; δω =

π

2 2

γ d

γ L

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 2

ωb3/2t1/2

6

TAE modes in MAST

(Culham Laboratory, U. K. courtesy of Mikhail Gryaznevich)

IFS numerical simulation Petviashvili [Phys. Lett. (1998)]

L linear growth without dissipation; for spontaneous hole formation; L d.

ω =(ekE/m)1/2 0.5L

With geometry and energeticparticle distribution known internalperturbing fields can be inferred

Predicted Nonlinear Frequency Sweeping Observed in Experiment

7

Study of Adiabatic EquationsStudy begins by creating a fully formed phase spacestructure (hole) at an initial time, and propagate solutionusing equations below.

€

∂f (J, t)

∂t−ν eff

3 ∂

∂J(∂J

∂EJ

∂f (J, t)

∂J) = 0 (in trapped particle region),

δωωb2 =

4γ L

π 2 ∂f0(ω0)

∂ω

dE[ f (J (E), t) − f0−ω b2

ω b2

∫ (ω0 +δω)]dφ

[2(E + ωb2 cosφ)]1/ 20

φ max

∫

γ dωb4 = −

γ L

dδω

dt

π∂f0 (ω0 )

∂ω

dJ[ f (J, t) − f00

Jsep

∫ (ω0 + δω)]

J =dϕ

2π∫ p =2

πdφ

0

φmac

∫ [2(E(J) +ωb2 cosφ)]1/2, E ≡ energy in local wave frame

Note: If ν eff = 0, ωb(δω, t) depends only on δω(t)

8

Results of Fokker-Planck Code

sweeping terminates why?

sweeping goes tocompletion

€

δ ˆ ω ≡δων eff

3/ 2

ωbi(γ dγ L2 )1/ 2

9

Normalized Adiabatic Equation, eff=0Dimensionless variables:

€

δωδω0

→ Ω, ωbi → 1, ωb → Ωb , Jsep → Ω b1, J → Ωb I, f (J) → GT (Ωb I)

€

Ωb =1− 1

ΩdIGT(IΩb)Q(I)

0

1

∫

I = (2 2)−1 dφ[E'+cosφ]1/20ϕ max

∫ , Q(I) =3 dφcosφ /[E'+cosφ]1/2

0ϕ max

∫dφ /[E' +cosφ]1/2

0ϕ max

∫

“BGK” Equation

Take derivative with with respect to Ωb

10

Propagation Equation;Difficulties

€

dΩdΩb

=ΩHT (Ω,Ω b)

1− Ω b

€

HT(Ω,Ω b ) = 1+ (ΩΩ b )−1 dIIdGT (IΩ b )

dI0

1

∫ Q (I)

Problems with propagation

a. HT (Ω Ω ) = 0, termination of frequency sweepingb. 1- Ω = HT (Ω Ω ) = 0; singularity in equation, unique solution cannot be obtained

11

Instability AnalysisBasic equation for evolving potential in frame of nonlinear wave (extrinsic wave damping neglected), 1= P(t) cos x + Q(t) sinx; Ω Ω Ω

€

dQ

dt+ ΔΩP = −β dΓf cos(x)∫

dP

dt− ΔΩQ = β dΓ f sin(x∫ )

f satisfies Vlasov equation for:

€

f ( x,v, t) = F (E ) +δ ˆ f (E, x)exp(−iω t)

Spatial solutions are nearly even or odd

12

Analysis (continued)

F(J)-F0(Ω ) ΩGT

Ω ΩJ

Find equilibrium in wave frame:

€

E = (v − Ω2 ) / 2 − Ω b2 cos x, J = ΩbI ; solve for Ωb = 1− dIGT (IΩb )Q(I)

0

1

∫ Linearization:

€

φ=[δP cos x + δQsin x]exp(−iω t); lowest order δQ = 0 + ϑ (Ω b / ΔΩ)

Perturbed distribution function

€

δf (E, x) = −∂GT (E )

∂EδPe −iωt[cosx − iω dt'e− iωt ' cos x(t ')( )

−∞

0

∫ ]

cos x(t)( ) = < cos x >2n cos4πn

T (E )t −τ (E, x)( )

⎡

⎣ ⎢

⎤

⎦ ⎥

n=1

∞

∑

13

Dispersion Relation

€

HT'

2≡ 1− β dE

∂GT (E)∂E−Ω b

2

Ω b2

∫ T (E )[< cos2 x > 0 − < cos x > 02]

= 2β dE-Ω b

2

Ω b2

∫n=1

∞

∑∂GT (E)

∂ET (E )

ω2 < cos x > 2n2

2n2π

T (E) ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

−ω 2

ω <<Ω b ⏐ → ⏐ ⏐ σΩ2, σ > 0

Identity

€

Instability Arises if H T' < 0

€

HT' = HT =1 +

1Ωb

dIIdG(Ωb I)

dI0

1

∫ Q(I)

Consequence: Adiabatic SweepingTheory “knows”about linear instability criterion for both types of Breakdown: (a)sweeping termination (b) singular point

Onset of instability necessitates non-adiabatic response

14

15

Comparison of Adiabatic Code and Simulation

€

Ωb0 =1.16, γ L

ωbi

=1.85, γ d

ωbi

= .093, I* = 0.8, ΔΩ

ωbi

= 9.26

(passing particle distribution flat)

16

Evolution of Instability

Trapping frequency,ωb ωbiSpectral Evolution, δωL

slope in passing particle distribution

Indication that Instability Leads to Sideband Formation

17

Side Band Formation During Sweeping

18

Summary1. Ideal model of evolution of phase structure has beentreated more realistically based on either particle adiabatic invariance or Fokker-Planck equation 2. Under many conditions the adiabatic evolution of frequency sweeping reaches a point where the theory cannot make a prediction (termination of frequency sweeping or singularity in evolution equation)3. Linear analysis predicts that these “troublesome” points are just where non-adiabatic instability arises4. Hole structure recovers after instability; frequency sweeping continues at somewhat reduced sweeping rate5. Indication the instability causes generation of side-bandstructures

19

Finis

20

Linear Dispersion Relation

€

HT(Ω,Ω b ) / 2 = −σ γ 2, if γ 2 << Ω b2 , σ > 0

Linear Instability if HT < 0

Hence HT(ΩΩb) =0 is marginal stability conditionof linear theory. Adiabatic theory breakdown due tofrequency sweeping termination, or reaching singularpoint is indicative of instability. Then there is an intrinsic non-adiabatic response of this particle-wave system

21